\[ \int \frac {x^2}{\left (b x^2+c x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {\arctanh \left (\frac {x \sqrt {b}}{\sqrt {c \,x^{4}+b \,x^{2}}}\right )}{b^{\frac {3}{2}}}+\frac {x}{b \sqrt {c \,x^{4}+b \,x^{2}}} \]
command
integrate(x^2/(c*x^4+b*x^2)^(3/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ -\frac {{\left (\sqrt {b} \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) + \sqrt {-b}\right )} \mathrm {sgn}\left (x\right )}{\sqrt {-b} b^{\frac {3}{2}}} + \frac {\arctan \left (\frac {\sqrt {c x^{2} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b \mathrm {sgn}\left (x\right )} + \frac {1}{\sqrt {c x^{2} + b} b \mathrm {sgn}\left (x\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \text {Exception raised: NotImplementedError} \]________________________________________________________________________________________